Universal logarithmic formulas for Segre classes on curve Quot schemes

Determine whether the generating series of Segre pushforwards for tautological bundles on relative curve Quot schemes admits a universal logarithmic expression in the relative classes \(\kappa_{a,b}=\pi_\star(c_1(L)^a c_1(\omega_\pi)^b)\), and characterize its universal power series by a closed recursion extending the recursion for relative symmetric products.

Background

The paper determines universal coefficients and recursive formulas for total Segre pushforwards of tautological line bundles on relative symmetric products C[n]=Sym⁡Bn(C)C^{[n]}=\operatorname{Sym}^n_B(C). This resolves the rank-one, one-insertion case of a question of Oprea–Pandharipande.

The unresolved extension replaces relative symmetric products by relative Quot schemes Quot⁡C/B(OC⊕N,n)\operatorname{Quot}_{C/B}(O_C^{\oplus N},n), for arbitrary N≥1N\ge 1. The question asks whether the same kind of universality and recursive structure persists in this broader setting.

References

Motivated by their study of Quot schemes, we ask the following question.

For $N\ge1$, let $\pi:C\to B$ be a smooth projective curve family, let $L$ be a line bundle on $C$, and let $\rho_n:\mathfrak Q_n=\operatorname{Quot}{C/B}(O_C{\oplus N},n)\to B$. Denote by $L{\mathfrak Q}{[n]}$ the tautological bundle on $\mathfrak Q_n$. Does

\sum_{n\ge0}qn(\rho_n)_\star s\bigl(L_{\mathfrak Q}{[n]}\bigr)

admit a universal logarithmic formula in the classes $\kappa_{a,b}=\pi_\star(c_1(L)a c_1(\omega_\pi)b)$? Can its universal series be characterized by a closed recursion extending Theorem \ref{segre_curve}?

— Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces  (2608.25447 - Kong, 26 Aug 2026) in Question 1.1, immediately after Theorem 1.2 in Section 1

It is natural to ask whether this open-locus modular interpretation admits a compactified extension.

For each $n$, let $\overline{\tau}n: [\overline{\mathcal M}{g,A_n}/S_n] \longrightarrow \overline{\mathcal M}g$ be the morphism forgetting the weighted markings and stabilizing. Let $\overline p_n:\overline{\mathcal U}_n \longrightarrow [\overline{\mathcal M}{g,A_n}/S_n]$ be the universal curve, let $\overline{\mathcal D}n\subset\overline{\mathcal U}_n$ be the universal unordered divisor of the weighted markings, and set $\overline{\mathbb V}_n=(\overline p_n)\star O_{\overline{\mathcal D}_n}$. This defines a compactified Segre series

$\overline{\mathsf Z}{\mathrm{Has}_g(q)

\sum_{n\ge0}qn\,(\overline{\tau}n)\star s(\overline{\mathbb V}_n) \in A*(\overline{\mathcal M}_g)_Q[[q]]$.

Can the compactified series

\overline{\mathsf Z}{\mathrm{Has}_g(q)

\sum_{n\ge0}qn\,(\overline{\tau}n)\star s(\overline{\mathbb V}_n)

be computed concretely in the tautological ring $R*(\overline{\mathcal M}_g)_Q[[q]]$?

— Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces  (2608.25447 - Kong, 26 Aug 2026) in Question 3.1, Section 3.2 (subsection “Hassett spaces”)